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Comment on “multistability, intermittency and remerging feigenbaum trees in an externally pumped ring cavity laser system”
Volume 116, number 7
PHYSICS LETTERS A
30 June 1986
COMMENT ON " M U L T I S T A B I L I T Y , I N T E R M r I T E N C Y AND R E M E R G I N G F E I G E N B A U M T R E E S IN AN EXTERNALLY P U M P E D R I N G CAVITY LASER S Y S T E M " Didier D A N G O I S S E and Pierre G L O R I E U X Laboratoire de Spectroscopie Hertzienne, associb au CNRS, Universitb de Lille L 59655 Villeneuve d'Ascq Cedex, France
Received 20 February 1986; accepted for publication 28 April 1986
Heffernan's bifurcation diagrams are shown to be scrambled by dynamic effects most probably due to a lack of convergence in his numerical calculations. Many of his conclusions have to be rejected.
In a recent letter [1] Heffernan claims he observed intermittency and multistability in a model of purely absorptive optical bistability. In that work, bifurcation diagrams for different values of the bistability parameter A (respectively input field Q) with the input field Q (respectively bistability parameter A) as control parameter are given. Although the author's interest is in " w h a t happens after many cavity lifetimes", we believe that these diagrams do not correspond to the actual limit points of the recurrent series which is used as a model of the Bloch-Maxwell equations in this case, namely Xn+ 1 Q - A x n / ( 1 + x2). This statement is based on the following: (i) the limit points of this series should be symmetric in the operation Q ~ - Q and x --* - x while the diagrams of figs. 4 and 5 of ref. [1] are not. (ii) The dissymetries present in these figures are characteristic of the dynamic deformation of a bifurcation diagram st~ch as what is observed even at very low "speed" when the control parameter is a function of time e.g. Q = Qo + at. When the rate a of variation of this control parameter is too high, the bifurcation points are shifted i.e. they are obtained at larger (respectively smaller) value than in the corresponding steady state diagrams when Q is increased (respectively decreased). The shape of the bifurcation diagram is also altered by dynamical effects. For instance, =
the first bifurcation from T to 2T periodic regime appears as a steep increase with an overshot when going from the T to the 2T region while it is softened with respect to the steady state at the 2T to T bifurcation. Dynamic deformations of bifurcation diagrams were reported recently by Kapral and Mandel for recurrent series [2] and by Midavaine et al. [3] and Brun et al. [4] for lasers. To support our interpretation of the diagrams of ref. [1] let us for example focus our attention on fig. 5 of ref. [1]. We have calculated the limits of the series used by Heffernan making sure that the displayed points are close to the asymptotic limit of the series within 10 -5 . The bifurcation diagrams obtained in this case reproduce well those obtained by Bier and Bountis [5] in the same case and the symmetry Q--. - Q , x ~ - x holds. On the other hand when the limit points are calculated assuming a quasi-continuity i.e. by using as initial value for x , ( Q ) the limit obtained for xn( Q -AQ) and iterating only once, the diagrams of Heffernan are reobtained when AQ is equal to 1 / 7 5 as illustrated in fig. 1. This figure reproduces bifurcation diagrams in both cases. Fig. l a has been obtained after checking the asymptotic value(s) while for fig. l b the quasi-continuity assumption as discussed above has been used. They clearly show that in the results of ref. [1] the limit points of the series were not reached and that this induces deformation or scrambling of the bifurca-
0375-9601/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
311
Volume 116, number 7
PHYSICS LETTERS A
tion d i a g r a m s similar to what is o b s e r v e d in the d y n a m i c regime. Fig. 1 has been given as an illustration of our statement. T h e s a m e holds for all the o t h e r bifurc a t i o n d i a g r a m s of ref. [1] (exception in fig. 3) which we have r e c a l c u l a t e d using the actual limit points. In the real d i a g r a m s , b u b b l e s a n d inverse F e i g e n b a u m trees a p p e a r as d e m o n s t r a t e d in ref.
x
2
/.~
H LL
/,.
H ~Z r CIZ
1
F-Z
-3
-'2
-~1 --INPUT
0 ~T ,
i
I
1 0
2
al 3
x2
30 June 1986
[5] b u t some of t h e m are usually " p a s s e d over" or s c r a m b l e d because of d y n a m i c a l effects. A s e m a n t i c p o i n t has also to be clarified. Heffernan claims that n o chaos is o b s e r v e d in this m a p p i n g since it is always fully deterministic. W e t h i n k that there has been now a general accept a n c e of the c o n c e p t of d e t e r m i n i s t i c chaos in m a n y fields of physics and chemistry [6] a n d H e f f e r n a n ' s s t a t e m e n t deserves some e x p l a n a t i o n since what is o b s e r v e d in his w o r k is a l m o s t exactly the chaos t h r o u g h a cascade of p e r i o d - d o u b l i n g b i f u r c a t i o n s which was one of the first clear insights into the routes leading to deterministic chaos [7]. Moreover, assigning a recurrent series to a differential e q u a t i o n requires some care if one w a n t s to get the same stability criteria for both. F o r instance the stability c o n d i t i o n of the fixed p o i n t s of xn+ 1 = Q - A x J ( 1 + x2~) of H e f f e r n a n differs f r o m that of the solution of d x / d r = Q - x A x / ( 1 + x2). To take care of this we could use for i n s t a n c e x , + 1 = Q / A - x , / A + x 3 / ( 1 + x~,) which has been o b t a i n e d b y rescaling the time by a factor of A. This series has the same fixed p o i n t s a n d stability c o n d i t i o n s as the a b o v e differential e q u a t i o n at least in the physically interesting region when A >t 1.
41 H LL
References
H
Z FA- 2
-Z:
b -'2
-'i INPUT
0~ FIELD
I~
2'
3
O
Fig. 1. Bifurcation diagrams of the series x,,+l = Q - Ax,/(1 + x 2) using (a) a check of the convergence at every point and (b) a quasi-continuity assumption. (b) reproduces well the results by Heffernan.
312
[1] D. Heffernan, Phys. Lett. A 108 (1985) 413; 109 (1985) 465 (E). [2] R. Kapral and P. Mandel, Phys. Rev. A 32 (1985) 1076. [3] T. Midavaine, D. Dangoisse and P. Glorieux, Phys. Rev. Lett. 55 (1985) 1989. [4] E. Brun, B. Derighetti, D. Meier, R. Holzner and M. Ravani, J. Opt. Soc. Am. B 2 (1985) 156. [5] M. Bier and T. Bountis, Phys. Lett. A 104 (1984) 239. [6] H. Haken, Synergetics (Springer, Berlin, 1978). [7] M.J. Feigenbaum, J. Stat. Phys. 19 (1978) 25; C. Tresser and P. Coullet, C.R. Acad. Sci. A 287 (1978) 577.
PHYSICS LETTERS A
30 June 1986
COMMENT ON " M U L T I S T A B I L I T Y , I N T E R M r I T E N C Y AND R E M E R G I N G F E I G E N B A U M T R E E S IN AN EXTERNALLY P U M P E D R I N G CAVITY LASER S Y S T E M " Didier D A N G O I S S E and Pierre G L O R I E U X Laboratoire de Spectroscopie Hertzienne, associb au CNRS, Universitb de Lille L 59655 Villeneuve d'Ascq Cedex, France
Received 20 February 1986; accepted for publication 28 April 1986
Heffernan's bifurcation diagrams are shown to be scrambled by dynamic effects most probably due to a lack of convergence in his numerical calculations. Many of his conclusions have to be rejected.
In a recent letter [1] Heffernan claims he observed intermittency and multistability in a model of purely absorptive optical bistability. In that work, bifurcation diagrams for different values of the bistability parameter A (respectively input field Q) with the input field Q (respectively bistability parameter A) as control parameter are given. Although the author's interest is in " w h a t happens after many cavity lifetimes", we believe that these diagrams do not correspond to the actual limit points of the recurrent series which is used as a model of the Bloch-Maxwell equations in this case, namely Xn+ 1 Q - A x n / ( 1 + x2). This statement is based on the following: (i) the limit points of this series should be symmetric in the operation Q ~ - Q and x --* - x while the diagrams of figs. 4 and 5 of ref. [1] are not. (ii) The dissymetries present in these figures are characteristic of the dynamic deformation of a bifurcation diagram st~ch as what is observed even at very low "speed" when the control parameter is a function of time e.g. Q = Qo + at. When the rate a of variation of this control parameter is too high, the bifurcation points are shifted i.e. they are obtained at larger (respectively smaller) value than in the corresponding steady state diagrams when Q is increased (respectively decreased). The shape of the bifurcation diagram is also altered by dynamical effects. For instance, =
the first bifurcation from T to 2T periodic regime appears as a steep increase with an overshot when going from the T to the 2T region while it is softened with respect to the steady state at the 2T to T bifurcation. Dynamic deformations of bifurcation diagrams were reported recently by Kapral and Mandel for recurrent series [2] and by Midavaine et al. [3] and Brun et al. [4] for lasers. To support our interpretation of the diagrams of ref. [1] let us for example focus our attention on fig. 5 of ref. [1]. We have calculated the limits of the series used by Heffernan making sure that the displayed points are close to the asymptotic limit of the series within 10 -5 . The bifurcation diagrams obtained in this case reproduce well those obtained by Bier and Bountis [5] in the same case and the symmetry Q--. - Q , x ~ - x holds. On the other hand when the limit points are calculated assuming a quasi-continuity i.e. by using as initial value for x , ( Q ) the limit obtained for xn( Q -AQ) and iterating only once, the diagrams of Heffernan are reobtained when AQ is equal to 1 / 7 5 as illustrated in fig. 1. This figure reproduces bifurcation diagrams in both cases. Fig. l a has been obtained after checking the asymptotic value(s) while for fig. l b the quasi-continuity assumption as discussed above has been used. They clearly show that in the results of ref. [1] the limit points of the series were not reached and that this induces deformation or scrambling of the bifurca-
0375-9601/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
311
Volume 116, number 7
PHYSICS LETTERS A
tion d i a g r a m s similar to what is o b s e r v e d in the d y n a m i c regime. Fig. 1 has been given as an illustration of our statement. T h e s a m e holds for all the o t h e r bifurc a t i o n d i a g r a m s of ref. [1] (exception in fig. 3) which we have r e c a l c u l a t e d using the actual limit points. In the real d i a g r a m s , b u b b l e s a n d inverse F e i g e n b a u m trees a p p e a r as d e m o n s t r a t e d in ref.
x
2
/.~
H LL
/,.
H ~Z r CIZ
1
F-Z
-3
-'2
-~1 --INPUT
0 ~T ,
i
I
1 0
2
al 3
x2
30 June 1986
[5] b u t some of t h e m are usually " p a s s e d over" or s c r a m b l e d because of d y n a m i c a l effects. A s e m a n t i c p o i n t has also to be clarified. Heffernan claims that n o chaos is o b s e r v e d in this m a p p i n g since it is always fully deterministic. W e t h i n k that there has been now a general accept a n c e of the c o n c e p t of d e t e r m i n i s t i c chaos in m a n y fields of physics and chemistry [6] a n d H e f f e r n a n ' s s t a t e m e n t deserves some e x p l a n a t i o n since what is o b s e r v e d in his w o r k is a l m o s t exactly the chaos t h r o u g h a cascade of p e r i o d - d o u b l i n g b i f u r c a t i o n s which was one of the first clear insights into the routes leading to deterministic chaos [7]. Moreover, assigning a recurrent series to a differential e q u a t i o n requires some care if one w a n t s to get the same stability criteria for both. F o r instance the stability c o n d i t i o n of the fixed p o i n t s of xn+ 1 = Q - A x J ( 1 + x2~) of H e f f e r n a n differs f r o m that of the solution of d x / d r = Q - x A x / ( 1 + x2). To take care of this we could use for i n s t a n c e x , + 1 = Q / A - x , / A + x 3 / ( 1 + x~,) which has been o b t a i n e d b y rescaling the time by a factor of A. This series has the same fixed p o i n t s a n d stability c o n d i t i o n s as the a b o v e differential e q u a t i o n at least in the physically interesting region when A >t 1.
41 H LL
References
H
Z FA- 2
-Z:
b -'2
-'i INPUT
0~ FIELD
I~
2'
3
O
Fig. 1. Bifurcation diagrams of the series x,,+l = Q - Ax,/(1 + x 2) using (a) a check of the convergence at every point and (b) a quasi-continuity assumption. (b) reproduces well the results by Heffernan.
312
[1] D. Heffernan, Phys. Lett. A 108 (1985) 413; 109 (1985) 465 (E). [2] R. Kapral and P. Mandel, Phys. Rev. A 32 (1985) 1076. [3] T. Midavaine, D. Dangoisse and P. Glorieux, Phys. Rev. Lett. 55 (1985) 1989. [4] E. Brun, B. Derighetti, D. Meier, R. Holzner and M. Ravani, J. Opt. Soc. Am. B 2 (1985) 156. [5] M. Bier and T. Bountis, Phys. Lett. A 104 (1984) 239. [6] H. Haken, Synergetics (Springer, Berlin, 1978). [7] M.J. Feigenbaum, J. Stat. Phys. 19 (1978) 25; C. Tresser and P. Coullet, C.R. Acad. Sci. A 287 (1978) 577.